3.3 The population of normal 3 The Galactic Pulsar Population3.1 Selection effects in pulsar

3.2 Correcting the observed pulsar sample 

3.2.1 Scale factor determination 

Now that we have a flavour for the variety and severity of the selection effects that plague the observed sample of pulsars, how do we decouple these effects to form a less biased picture of the true population of objects? A very useful technique, first employed by Phinney & Blandford and Vivekanand & Narayan [191, 259], is to define a scaling factor tex2html_wrap_inline9277 as the ratio of the total Galactic volume weighted by pulsar density to the volume in which a pulsar could be detected by the surveys:


In this expression, tex2html_wrap_inline9279 is the assumed pulsar distribution in terms of galactocentric radius R and height above the Galactic plane z . Note that tex2html_wrap_inline9277 is primarily a function of period P and luminosity L such that short period/low-luminosity pulsars have smaller detectable volumes and therefore higher tex2html_wrap_inline9277 values than their long period/high-luminosity counterparts. This approach is similar to the classic tex2html_wrap_inline9293 technique first used to correct observationally-biased samples of quasars [211].

This technique can be used to estimate the total number of active pulsars in the Galaxy. In practice, this is achieved by calculating tex2html_wrap_inline9277 for each pulsar separately using a Monte Carlo simulation to model the volume of the Galaxy probed by the major surveys [170]. For a sample of tex2html_wrap_inline8995 observed pulsars above a minimum luminosity tex2html_wrap_inline9299, the total number of pulsars in the Galaxy with luminosities above this value is simply


where f is the model-dependent ``beaming fraction'' discussed below in §  3.2.3 . Monte Carlo simulations of the pulsar population incorporating the aforementioned selection effects have shown this method to be reliable, as long as tex2html_wrap_inline8995 is reasonably large [131].

3.2.2 The small-number bias 

For small samples of observationally-selected objects, the detected sources are likely to be those with larger-than-average luminosities. The sum of the scale factors (Equation (5Popup Equation)), therefore, will tend to underestimate the true size of the population. This ``small-number bias'' was first pointed out by Kalogera et al. [112Jump To The Next Citation Point In The Article, 113Jump To The Next Citation Point In The Article] for the sample of double neutron star binaries where we know of only three clear-cut examples (§  3.4.1). Only when the number of sources in the sample gets past 10 or so does the sum of the scale factors become a good indicator of the true population size.


Click on thumbnail to view image

Figure 15: Small-number bias of the scale factor estimates derived from a synthetic population of sources where the true number of sources is known. Left: An edge-on view of a model Galactic source population. Right: The thick line shows tex2html_wrap_inline8993, the true number of objects in the model Galaxy, plotted against tex2html_wrap_inline8995, the number detected by a flux-limited survey. The thin solid line shows tex2html_wrap_inline8997, the median sum of the scale factors, as a function of tex2html_wrap_inline8995 from a large number of Monte-Carlo trials. Dashed lines show 25 and 75% percentiles of the tex2html_wrap_inline8997 distribution.

3.2.3 The beaming correction 

The ``beaming fraction'' f in Equation (5Popup Equation) is simply the fraction of tex2html_wrap_inline9317 steradians swept out by the radio beam during one rotation. Thus f gives the probability that the beam cuts the line-of-sight of an arbitrarily positioned observer. A naïve estimate of f is 20%; this assumes a beam width of tex2html_wrap_inline9323 and a randomly distributed inclination angle between the spin and magnetic axes [238]. Observational evidence suggests that shorter period pulsars have wider beams and therefore larger beaming fractions than their long-period counterparts [171Jump To The Next Citation Point In The Article, 149Jump To The Next Citation Point In The Article, 32Jump To The Next Citation Point In The Article, 231Jump To The Next Citation Point In The Article]. It must be said, however, that a consensus on the beaming fraction-period relation has yet to be reached. This is shown in Fig.  16 where we compare the period dependence of f as given by a number of models. Adopting the Lyne & Manchester model, pulsars with periods tex2html_wrap_inline9327 ms beam to about 30% of the sky compared to the Narayan & Vivekanand model in which pulsars with periods below 100 ms beam to the entire sky.


Click on thumbnail to view image

Figure 16: Beaming fraction plotted against pulse period for four different beaming models: Tauris & Manchester 1998 (TM88; [231]), Lyne & Manchester 1988 (LM88; [149]), Biggs 1990 (JDB90; [32Jump To The Next Citation Point In The Article]) and Narayan & Vivekanand 1983 (NV83; [171]).

When most of these beaming models were originally proposed, the sample of millisecond pulsars was tex2html_wrap_inline9329 5 and hence their predictions about the beaming fractions of short-period pulsars relied largely on extrapolations from the normal pulsars. A recent analysis of a large sample of millisecond pulsar profiles by Kramer et al. [122Jump To The Next Citation Point In The Article] suggests that the beaming fraction of millisecond pulsars lies between 50 and 100%.

3.3 The population of normal 3 The Galactic Pulsar Population3.1 Selection effects in pulsar

image Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de